Method for determining time constants of a reference model in a cascade controlling circuit

ABSTRACT

A method for determining at least one time constant of a reference model, which is designed as a 2nd order time-delay element of a machine. The method includes detecting an oscillation frequency of an undamped machine oscillation and determining an optimized value of a time constant of the reference model as a function of the detected oscillation frequency of the undamped machine oscillation.

Applicant claims, under 35 U.S.C. §§ 120 and 365, the benefit ofpriority of the filing date of Sep. 21, 2000 of a Patent CooperationTreaty patent application, copy attached, Ser. No. PCT/EP00/09232, filedon the aforementioned date, the entire contents of which areincorporated herein by reference, wherein Patent Cooperation Treatypatent application Ser. No. PCT/EP00/09232 was not published under PCTArticle 21(2) in English.

Applicant claims, under 35 U.S.C. § 119, the benefit of priority of thefiling date of Sep. 24, 1999 of a German patent application, copyattached, Ser. No. 199 45 748.4, filed on the aforementioned date, theentire contents of which are incorporated herein by reference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a method for determining at least onetime constant of a reference model in a cascaded controllingarrangement.

2. Description of the Related Art

Usually a cascaded controlling structure, including a position, rpm andcurrent control device, is employed in numerically controlled machinetools. As a rule, the speed control device, which is connecteddownstream of the position control device, is embodied as a PI speedcontrol device and includes a proportional branch (P) and an integralbranch (I). The phase response of the upstream connected positioncontrol device worsens as a result of the effect of the integral branchof the speed control device. It is therefore necessary as a consequenceof this to reduce the loop gain kV of the position control device apriori in order to prevent oscillations in the drive systems of themachine tool controlled by the controlling device. However, as large aspossible a loop gain kV of the position control device is desired inprinciple.

To solve these problems, it has already been suggested by P. Ernst andG. Heinemann in the course of a seminar presentation under the title“Optimierte Achsregelung mit durchgängig offenen CNC-Steuerungen”[Optimized Axis Control with Continuously Open CNC Controls] (ISWPosition Controlling Seminar 1999, 26, Mar. 27, 1999) in Chapter 2.2 toconnect a reference model upstream of the speed control device. Thereference model, designed as a 2nd order time-delay element, is matchedto the behavior of the closed speed control device without an integralportion. It is possible in this way to eliminate, or at least tominimize, the detrimental influence of the integral portion on thecontrol behavior of the speed control device. However, the desiredelimination of disturbances without integral portions continues to befully maintained. However, no further suggestions can be found in thecited reference regarding suitable parameterization, in particular thedetermination of suitable time constants, of a corresponding 2nd orderreference model.

SUMMARY AND OBJECTS OF THE INVENTION

It is therefore an object of the present invention to disclose a methodfor determining at least one time constant of a 2nd order referencemodel, which is arranged in a cascaded controlling device of a machinebetween a position control device and an speed control device, and whichassures an optimized control behavior of the controlling device.

This object is attained by a method for determining at least one timeconstant of a reference model, which is designed as a 2nd ordertime-delay element of a machine. The method includes detecting anoscillation frequency of an undamped machine oscillation and determiningan optimized value of a time constant of the reference model as afunction of the detected oscillation frequency of the undamped machineoscillation.

The parameterization of a suitable 2nd order reference model for themost varied types of machines is now possible by the method of thepresent invention. Here, the resulting reference model essentiallyalways assures that at least the undesired influence of the integralportion of the speed control device on the control behavior iseliminated.

It should be noted that the machine tools controlled in the past and bythe present invention can generally thought of as falling with one oftwo categories. One category or type of machine tool regards rigidmachines that are not too large in structural size, which is mostlydirectly driven or has linear motors. A second category or type ofmachine tool regards machine tools with a dominant natural frequency inthe range between 15 Hz to 80 Hz, in which no sufficiently large kVfactor can be set.

Depending on the machine type, one time constant or two time constantsare determined in accordance with the present invention, which determinethe behavior of the reference model and therefore affect the controlbehavior of the controlling arrangement during the actual controllingoperation. However, in accordance with the present invention at leastthe so-called second time constant of the reference model is basicallydetermined as a function of a detected oscillation frequency of acontinuous machine oscillation.

Surprisingly, or contrary to theoretical reflections, it is now possibleby the steps of the present invention for determining the time constantto also compensate controlled systems with idle times and delay elementsfor machines which theoretically would require higher order referencemodels; this applies in particular to the above mentioned category ofnon-rigid machines with dominant natural frequency. The determination oftheoretically exact nth-order reference models (n>2) in such machineswould be connected with a very large outlay. In contrast to this it ispossible by the use of second order time-delay elements as the referencemodel, whose time constants are determined in accordance with thepresent invention, to keep the resulting outlay for parameterization ofthe reference model low.

The method in accordance with the present invention can be performedmanually, as well as in an automated manner.

Further advantages, as well as details of the method in accordance withthe present invention ensue from the subsequent description of exemplaryembodiments by the attached drawings.

Shown here are in:

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a block diagram representation of a part of an embodimentof a cascaded controlling structure of a numerically controlled machinetool in accordance with the present invention;

FIGS. 2 a and 2 b show a flow diagram in each for explaining anembodiment of a method of the determination, in accordance with thepresent invention, of the time constant of a 2nd order reference modelto be used with the cascaded controlling structure of FIG. 1;

FIGS. 3 to 21, respectively different representations, which will beexplained in greater detail hereinafter.

DESCRIPTION OF THE PREFERRED EMBODIMENT(S) OF THE INVENTION

In a greatly schematized form, FIG. 1 shows a block diagramrepresentation of a part of a cascade controlling structure of anumerically controlled machine tool, such as is known, for example, in asimilar shape from the above discussed reference.

The portion of the controlling structure represented includes a positioncontrol device 10, as well as a downstream-connected speed controldevice 20. The actual controlled system 30 is arranged downstream of thespeed control device 20 and is only schematically indicated. In thepresent example, the speed control device is embodied as a PI controldevice (proportional-integral control device); the integral branch 21,as well as the proportional branch 22 of the speed control device 20 arerepresented separately of each other in FIG. 1. A reference model 40 isarranged between the position control device 10 and the speed controldevice and is embodied as a 2nd order time-delay element, i.e. aso-called PT2 element. The reference model 40 simulates the behavior ofthe closed speed control device 20 without an integral portion and inthis way assures that at least the undesired influence of the integralportion, or integral branch 21, on the control behavior of the speedcontrol device is eliminated. As already indicated above, by the stepsto be explained in what follows it is possible in a surprising manner toalso parameterize reference models which compensate controlled systemswith idle times and delay elements. In theory it would be necessary toparameterize reference models with orders n>2 for such controlledsystems, which would be relatively expensive.

The transfer function H(s) of the reference model 40 embodied as a 2ndorder time-delay element results in a known manner from the followingequation (1):H(s)=K/(1+T 1*s+(T 2)² *s ²)  Equ. (1)

The two time constants T1 and T2 are decisive for the layout, or theparameterization of the reference model 40. They must be determined as afunction of the respective machine, or of the controlling conditions.

Contrary to theoretical considerations it is shown by the presentinvention that the use of 2nd order reference models, whose timeconstants T1 and T2 are determined in accordance with the presentinvention, is even possible when the respective system would actuallyhave to be simulated by a reference model of higher order n, i.e. n>2.However, the mathematically exact representation of such a complexsystem by an appropriate nth order reference model would basically causean extremely high computational effort. In actuality this has the resultthat by the use of a 2nd order reference model whose time constants T1and T2 are determined by means of the invention, it is possible to alsooptimize the control behavior of the speed control device 20 formachines which are part of the second category already discussed above.Here, by employing a 2nd order reference model, which is parameterizedin accordance with the present invention, in these systems, not only isthe influence of the integral branch of the speed control deviceeliminated, but moreover the influence of additional delays, or idletimes, in the controlled system is also minimized. It is surprisinglypossible to use loop gains kV in such systems with 2nd order referencemodels parameterized in accordance with the present invention, which aregreater than possible loop gains kV in case of a non-existing, orswitched off integral branch in the speed control device.

The operation in accordance with the present invention for determiningthe time constants T1, T2 for the 2nd order reference model will now beexplained by the flow diagrams in FIGS. 2 a and 2 b.

In the first part of the method explained in what follows, representedin FIG. 2 a, first the time constant T2, or a correspondingly optimizedvalue T2_OPT of the second time constant T2 will be determined.

In a first method step S10, first the determination, or presetting, ofstarting values T1_0 ₁ and T2_0 ₁ for the first and second time constantT1, T2 takes place. In the present example, the starting values T1_0 ₁and T2_0 ₁ equaling T1_0 ₁=O and T2_0=0 are selected. This selection ofthe starting values T1_0 ₁ and T2_0 ₁ equaling T1_0 ₁=O and T2_0 ₁=0means for the entire system in the end that the 2nd order referencemodel is switched out of the controlling arrangement, or is not active.

In what follows, the loop gain kV of the position control device isincreased in steps in the following steps S20 and S30 and a check ismade after each increase to determine whether an oscillation in therespective machine is already recognizable. This takes place until at afirst maximum loop gain kV_(max1) an almost undamped oscillation of themachine at a defined oscillation frequency f_(S1) can be registered.

If an appropriate undamped oscillation of the machine can be registered,the associated oscillation frequency f_(S1) is measured, or determined,in accordance with the method step S40.

Thereafter, in the method step S50, the two optimized values T2_OPT andT1_OPT can be determined for the two time constants T1, T2. Here, theoptimized value T2_OPT for the second time constant T2 can be determinedas a function of the oscillation frequency f_(S1) determined in stepS40, i.e.T 2 _(—) OPT=f(f _(S1))=1/(2*π*f _(s1))  Equ. (2)

The optimized value T1_OPT for the first time constant T1 results frompredetermined system parameters in accordance with the followingequation:T 1 _(—) OPT=(J _(L)*2*π)/(k _(p) *K _(MC))  Equ. (3)wherein J_(L): Momentary load,

-   -   k_(p): Loop gain of the proportional branch of the speed control        device,    -   K_(MC): Motor constant.

Subsequently a check is performed in method steps S60 to S85 whether thepreviously determined time constants T1, T2 of the 2nd order referencemodel assure the desired control behavior during the controllingoperation. Moreover, a maximum loop gain kV of the position controldevice for the optimized time constants T1_OPT, T2_OPT is set in thesemethod steps.

For this purpose, initially a check is made in the method step S60whether an undamped oscillation of the machine results in the systemwhen using the previously determined optimized values T1_OPT, T2_OPT andthe first maximum loop gain kV_(max1) determined in step S30.

If this is not the case, the loop gain kV is increased by steps inmethod steps S70 and S80 until an undamped machine oscillation can beregistered at a loop gain kV_(max2). The maximum loop gain kV_(max2)determined in this way at which, in connection with the time constantsT1_OPT, T2_OPT of the 2nd order reference model, an undamped machineoscillation occurs, is subsequently multiplied by a safety factor K<1 inmethod step S85. From this then results the optimized loop gain kV_OPTfor the position control device, which can be used for a stable systemduring controlling operations, i.e.kV _(—) OPT=K*kV _(max2)  (Equ. 4)

The safety factor K can be selected as K=0.6, for example, in order toassure sufficient stability of the position control device in this way.

However, if it is found in method step S60 that, when using thepreviously optimized time constants T1_OPT, T2_OPT in the referencemodel and the loop gain kV_(max1), an undamped machine oscillationalready results at an oscillation frequency f_(S2), the oscillationfrequency f_(S2) is determined and an optimized time constant T2_OPT isagain determined in method step S65 as a function of the oscillationfrequency f_(S2) in accordance withT 2 _(—) OPT=f(f _(S2))=1/(2*π*f _(S2))  (Equ. 2′).

If required, the determination of an optimized time constant T2_OPT isrepeated several times in method steps S60 and S65, until finally noundamped machine oscillation can be registered at the selectedparameters of T1_OP, T2_OPT.

In connection with machines of the first category, the second orderreference model is basically parameterized after these steps, i.e. thetwo time constants T1 and T2 are determined. If it is intended tooptimize the control structure of a machine of the second category,further method steps are required for suitably determining the firsttime constant T1 of the reference model in particular. This will beexplained in what follows by means of FIG. 2 b.

It is of course also possible to perform the following steps fordetermining a suitable first time constant T1 even with the mentionedmachines of the first category in order to check in this way whether thevalue for T1_OPT set in accordance with the above Equ. (3) provides anacceptable system behavior.

Thus, for determining an optimized value T1_OPT′ of the first timeconstant T1, first a second start value T1_0 ₂ for the first timeconstant T1 is set in method step S90. For this, the value for T1determined in step S50 in accordance with Equ. (3) is used as the secondstart value T1_0 ₂, i.e. T1_0 ₂=T1_OPT.

Thereafter the first time constant T1 is changed in method step S100,for example increased, and a check is subsequently made in method stepS120 whether an undamped machine oscillation can already again beregistered. Besides the increase of the first time constant T1 in stepS100 it would basically also be conceivable that it be decreased.

As long as no undamped machine oscillation can be registered, the loopgain kV is increased in method steps S110 and S120 stepwise up to a loopgain kV_(max3), at which an undamped machine oscillation can beregistered.

A check is thereupon made in method step S130, whether the loop gainkV_(max3) determined in this way is greater than the loop gainkV_(max2), which had been maximal up to this time. If it is, then theamplification factor kV_(max3) is set equal to kV_(max2) per step S135and the method continues with step 100.

If the loop gain kV_(max3) is greater than the loop gain kV_(max2),which had been maximal up to this time, the loop gain kV_(max3) is setto equal kV_(max2), and a run-through of the method steps starting withS100 takes place again. This means that a check is made in the endwhether with a changed value for T1_OPT a higher value for the loop gainkV can possibly be set.

This takes place until in method step S130 it is determined that theloop gain kV_(max3) is no longer greater than the loop gain kV_(max2)determined during the previous run-through.

In accordance with method step S140, the value for the first timeconstant T1 then present, besides the already previously determinedvalue T2_OPT, is used as the optimized value T1_OPT for parameterizingthe 2nd order reference model.

Furthermore, similar to the procedure in FIG. 2 a, the last determinedmaximum loop gain kV_(max2) is multiplied by a correction factor K<1, inorder to again assure the stability of the position control device, i.e.the optimized value kV_OPT for the loop gain of the position controldevice again results askV _(—) OPT=K*kV _(max2)  Equ. (5)

Thus, besides the two determined parameters T1 and T2 for the 2nd orderreference model to be used, there is now also an optimized maximum loopgain kV_OPT for the position control device, which can be used in thesubsequent controlling operation.

Alternative forms of embodiment also exist within the scope of thepresent invention.

The theoretical considerations on which the present invention is basedwill be explained in greater detail in what follows in the followingdescription and several simulations and test results will be presented.

Theory, Simulation and Test Results

1. Simulation with a Simplified Controlling Model

1.1 Model of the Controlled System

The method of the present invention and the arrangement of the presentinvention were tested by a mathematical simulation. This simulationwhich, besides the mathematical machine model, also contains themathematical model of the present invention, will be described in whatfollows.

The mass inertia moment of the controlled system, together with themomentary constants of the motor, are the defining characteristics ofthe system. The following parameters are used in connection with this:

Mass inertia J_(L)=50 kgcm²

Motor constant k_(MC)=(1.5/2)*(Nm/A_(eff)), wherein A_(eff) is known inthe art to represent an effective motor current which is measured inAmperes

Therefore, the controlled system G(s) is determined by:G(s)=(num/den)=1/(J _(L) *s).

The conversion from the radian frequency ω to U/s (U represents thenumber of rotations) takes place by a downstream-connected P-elementwith 1/(2*π). A disturbance can be introduced via the input “momentarydisturbance Ms”, which simultaneously affects the momentary value andthe actual rpm. This is intended to correspond to a typical disturbancebecause of a milling cutter action and is used to rate the disturbancerigidity.

For simulating realistic rpm-connected losses, a derivative feedback k′pof the internal system output to the momentary summing point takesplace. By this a new controlled system G′(s) is created:G′(s)=(1/(J _(L) *s))/(1+k _(p)′/(J _(L) *s)))G′(s)=(1/(k _(p)′+(J _(L) *s))G′(s)=1/k _(p)′*1/(1+(J _(L) /k _(p) ′*s))

A TP1 control device is created by this derivative feedback.

A model of the 1^(st) order controlled system with disturbanceintroduction is represented in FIG. 3. As shown in FIG. 3, a signal 1 qis fed to an amplifier 300 that multiplies the signal 1 q by a momentaryconstant to generate a signal 302 that is fed to adder 304 A momentarydisturbance signal Ms is fed to the adder 304. As shown in FIG. 3, theadder 304 is connected to a control system 306 that generates the signalG(s)=(num/den) which is fed to a component 308 that generates a losssignal 310 that is a function of rpm. The loss signal 310 is fed back tothe adder 304. The signal G(s) and the signal Ms are each fed to asecond adder 312 that adds the two signals to generate signal 314. Thesignal 314 is then fed to an amplifier 316 to generate signal Msl.

1.2 Model of the Disturbed Controlled System

A model of the disturbed controlled system is represented in schematicform in FIG. 4. The controlled system is charged with a disturbancepulse of 2 Nm and of a length of 70 ms. The start time lies at 40 ms.

As shown in FIG. 4, the signal 1 q is fed to a control system 400 withcontrolled disturbance that is supplied by a momentary disturbancedevice 402. The control system generates a resultant signal nsi.

This disturbed controlled system is integrated into a simulation as agroup “disturbed controlled system A->U/s”.

1.3 Simulation Model

The simulation model contains a closed position control device loop. Foralignment purposes of the speed control device it is possible tointroduce a skip of 200 mm/min to the speed control device via a switch1. A suitable simulation model for examining a 1st order reference modelis represented in FIG. 5. The IPC reference model can be switched on andoff with the switch upstream of Sum 1.

As shown in FIG. 5, a jerk signal 500 is fed to a switch 502 along withan rpm jerk signal 504 and a feedback signal 506. The switch 502 sendsone of the three signals 500, 504, 506 to an amplifier 508 where thesignal is multiplied to generate a resultant signal 510. The resultantsignal 510 is sent to an adder 512, a reference model component 514 anda second switch 516. At the reference model component 514, the resultantsignal 510 is operated by the factor (2)/MP2500 and the signal 518 isfed to the switch 516. The switch 516 selects one of the signals 518,510 and a reference model signal 520. The selected signal is sent to anadder 522 that adds the selected signal with a disturbance signal 524.The combined signal 526 is fed to component 528 that applies the factorMP2510/s to generate signal 530 that is fed to adder 532 and multiplexer534.

At the adder 512, the resultant signal 510 is added to the disturbancesignal 524 to generate a signal 536 that is fed to amplifier 538 thatmultiplies the signal 536 by a factor that results in signal 540 that islater fed to adder 532 and multiplexer 534. The adder 532 generates asignal 541 that is fed to control system 542 that adjusts the signal 541to take into account disturbance effects. The signal 524 output from thecontrol system 542 is fed back to both adder 512 and adder 522. Thesignal 524 is also fed to an amplifier 546 that generates a signal 548that is fed to multiplexer 550 and component 552 that applies the factor1/s. The signal 554 generated by component 552 is fed to amplifier 556and the amplified signal 558 is fed to adder 560.

A pulse generator 562 generates a signal that is operated by a point setpoint component 564 that applies a factor 1/s to generate signal 566.The signal 566 is fed to both adder 560, adder 568 and multiplexer 570.The adder 568 adds the signals 558 and 566 to generate signal 572 thatis multiplied in amplifier 574 and the signal 576 is fed to multiplexer550. The adder 560 adds signals 558 and 566 to generate a signal that isamplified by amplifier 577 to generate feedback signal 506.

The multiplexer 570 generates a signal 578 that is fed to amplifier 580and the signal 582 is fed to multiplexer 550. The multiplexer 550 sendsits signal to a multiplexer 584 that also receives a signal frommultiplexer 534. The multiplexer 584 generates a signal 586 that isreceived by component 588 that is a MATLAB data file where all resultingsimulation data results are stored and from which all graphs shown inFIGS. 6-12, 14-17 and 20 are extracted.

2. Determination of the Simulation Parameters

It was necessary to determine the control device amplifications forparameterizing the control devices.

2.1 Alignment of the Speed Control Device

For the alignment of the speed control device the disturbance moment ofthe controlled system was temporarily set to 0, and the switch 1 was setto skip. The skip size was 200

Once the skip size is set to 200 mm/min the kinematics and current flowof the system are represented by the graphs of FIG. 6. In particular,the top graph represents the position (mm)/velocity (mm/s) of the systemas a function of time (s). The curve sactual represents the actualposition, snominal the nominal position, sdiff is snominal−sactual andvactual is the actual velocity. The lower graph maps the variouscurrents (A) of the system versus time (s). The curve 1(ki) representsthe current of the integral branch motor current, the curve 1(kp)represents the proportional branch motor current and 1 q=1(ki)+1(kp).Similar graphs are presented in FIGS. 7-10, 14-17 and 20. One differencein the graphs is that the graphs of FIGS. 14-17 and 20 is that the timescale is in minutes.

The conditions represented in FIG. 6 resulted for the loop gains for

P-factor (speed control device)=9

I-factor (speed control device)=2200 of the control device.

The simulation results correspond to a real drive mechanism. The controlstar time was set as Ta=4.6 ms.

2.2 Determination of the Position Control Device Amplification kV

To determine the maximum position control device amplification, theI-portion of the speed control device was set to 0.

Position control device amplification=15

P-factor (speed control device)=9

I-factor (speed control device)=0

The kV factor was set such that no oscillation of the actual motorcurrent Iq occurred.

The low disturbance rigidity without the I-portion can be seen from thecontour variation curve in FIG. 7. No complete removal of thedisturbance takes place.

2.3 Activating the I-Portion of the Speed Control Device

The I-portion of the speed control device was activated without theposition control device amplification being reduced.

Position control device amplification=15

P-factor (speed control device)=9

I-factor (speed control device)=2200

In accordance with FIG. 8 it can be easily seen from the motor currentsthat the system oscillates. The kV factor (or the I-portion of the speedcontrol device) must be reduced.

2.4 Reduction of the Position Control Device Amplification

The kV factor of the position control device was reduced until there wasno longer a tendency to oscillate.

Position control device amplification=9

P-factor (speed control device)=9

I-factor (speed control device)=2200

The contour variation increases (bad control behavior) because of thesmaller kV factor, but the disturbance rigidity is improved incomparison with a system without an I-portion (see FIG. 9).

2.5 Series-Connection of the IPC Reference Model (1st Order) with theI-Portion

The kV factor, which in the beginning had been possible without theI-portion of the speed control device, was set. In addition, theI-portion of the aligned speed control device was set. The referencemodel was realized in the 1st order (neglecting the derivative lossfeedback of the controlled system).

Position control device amplification MP1510 [m/min/mm]=15 P-factor(speed control device)

MP2500 [As]=9 I-factor (speed control device) MP2510 [A]=2200, wherein Arepresents Amperes.

It is possible to read out of the diagram in FIG. 10, that with a lowcontour variation a large disturbance rigidity is provided.

3. Calculation of the IPC Reference Model

The basis for the reference model is that all portions of the P controldevice, including the system, do not reach the integrator. Therefore asimplified model of the closed control loop (only the P control deviceis active) was inserted into the set point default of the integrator.The motor losses are not considered.

3.1 Calculation from Model Parameters

The following physical values appear in this closes control loop:

P-factor speed control device: in [As/U]

Motor constant: kMC/sqrt(2) in [Nm/A]

Moment of mass inertia of the system J_(L)

Thus, the conversion function G(s) of the open control loop is:G(s)=MP 2500*k _(MC)*1/(2*π)*1/(J _(L) *s) k _(p) ′=MP 2500*k _(MC)*1/(2*π)G(s)=k _(p)′*1/([J _(I) ]J _(L) *s)

The conversion function H(s) of the closed control loop is:H(s)=G(s)/(1+G(s))H(s)=(k _(p)′/(J _(L) *s))/(1+(k _(p)′/(J _(L) ]*s)))H(s)=1/(1+(J _(L) *s)/k _(p)′)H(s)=1/(1+T ₁ *s)

A PT1 element with the time constant T1 is obtained as the IPC referencemodel:T ₁ =J _(L) /k _(p)′=(J _(L)*2*π)/(MP 2500*k _(MC))  (F1)3.2 Calculation from Machine Parameters

Heidenhain controls have an acceleration feedforward control, which canbe set by a machine parameter. This machine parameter MP26 provides thereciprocal value of the angular acceleration a per current in [As²/U].The time constant of the IPC can be calculated in a simple manner by theangular acceleration.

Mel=Electrical moment [Nm]

kMC=Momentary motor constant [Nm/A]

J_(L)=Moment of mass inertia [kgm²]

MP26=Acceleration feedforward control [As²/U]M _(el)=I_(MOT) *k _(MC)α=M _(el) /J _(L) α=(I _(MOT)*2*π)MP 26

This is equal to:J _(L) /k _(MC)=(MP 26)/2*π

This inserted in (F1):T ₁ =J _(L) /k _(p)′=(J _(L)*2*π)/(MP 25*k _(MC))T ₁ =MP 26/MP 25  (F2).

Although the IPC should be assigned to the integral factor of the speedcontrol device, the IPC-MP should be among the feedforward controlparameters, since it can only be used after MP26 has been determined.

4. Examination of the Phase Response of the Speed Control Device Loop

To examine the phase response, the phase shift of the closed speedcontrol circuit is examined. A simulation model which contains, interalia, the set point and actual speed, is used for this. The followingphase responses were determined here.

4.1 Phase Response without IPC

The phase response without IPC is represented in FIG. 11. It can be seenthat a limit in the phase does not result sooner than at −180°. Areduction of the phase edge results because of the I-portion of thespeed control device, together with additional delays, idle times andlarge masses.

4.2 Phase Response with IPC

The phase response with IPC is represented in FIG. 12. With IPC thephase is only shifted by maximally −90°. Greater stability (or higherkV) of the position control device ensue because of the increase in thephase edge.

Please note that the graphs shown in FIGS. 11 and 12 are known as Bodediagrams which are used to characterize the behavior of a filter. Inparticular, the upper graphs of FIGS. 11 and 12 show the amplitude/gainresponse of the filter. The lower graphs show the phase response of thefilter. The x-axis represents the frequency over a certain range.

5. Consideration of the IPC in Feedforward Control

All previous reflections were made without feedforward controls (draggedoperation). In what follows, the feedforward control will be included.

For reasons of clarity, the speed control device in the simulation modelwas realized in its own block and was equipped with the following inputs(from top to bottom):

Switching the IPC on or off

Switching the feedforward control on or off

Acceleration feedforward control from the interpolator (IPO)

Speed feedforward control from the IPO

Set point rpm

Actual rpm.

The speed control block has the following outputs:

Three signals (via a multiplexer) for monitoring the currents in thespeed control device

Momentary current output Iq of the speed control device.

The structure of the position control device simulation with feedforwardcontrol is represented in FIG. 13. The speed feedforward control (Sum6)had additionally been integrated into the position control circuit.

By connecting the disturbance moment with the appropriate input of thecontrolled system, a disturbance can act as before on the controlledsystem.

The system allowance comes from the interpolator block (IPO). It ispossible to perform a parameterization of jerk, acceleration, speed anddistance via the Matlab dataset “M_IPO.M”. “M_IPO.M” is also called upwithin “M_IPC.M”.

As shown in FIG. 13, a jerk signal 600 is fed to a switch 602 along witha speed feedforward control signal 604 and an rpm jerk signal 606. Theswitch 602 sends one of the three signals to an amplifier 608, whichapplies MP2020 to generate a signal 610 that is fed to the speed controlsystem 612.

The speed control system 612 receives five other signals. One of thesignals 614 is generated by the IPC component 616 and another of thesignals is the jerk signal 600. The two other signals 622, 624 areinitiated by interpolator 626 where the a_soll and w_soll signals fromthe interpolator are amplified by amplifiers 628, 630, respectively,that apply M02020 to generate the resultant signals received by thecontrol system 612.

The control system 612 generates a signal 632 that is received by acontrol system with disturbance introduction 634 whose output signal 636is fed back to the control system 612 and a component 638 that applies1/s to the signal which results in signal 640. Signal 640 is fed to anamplifier 642 that applies MP2020 to generate a signal 644 that isreceived by adder 646 that also receives a signal s_soll. The adder 644combines the two signals to provide a signal 648 that is amplified viaamplifier 649 and received by adder 650. The adder 650 sums the signalfrom amplifier 649 with a signal 652 that is the result of theamplification, via amplifier 654, of signal a_soll generated by theinterpolator 626. The speed control signal 604 is then fed back to theswitch 602.

As shown in FIG. 13, the signal s_soll is also fed to an amplifier 656that sends the amplified signal to an adder 658 and a multiplexer 660.The adder 658 receives a signal 662 that is the result of theamplification of signal 644 via amplifier 664. The signal 662 is alsosent to multiplexer 660. The multiplexer 660 and the adder 658, incombination with amplifier 666, send signals 668, 670 to a multiplexer672. The multiplexer 672 also receives a signal 674 from an amplifier676 that amplifies signal 636.

The multiplexer 672 generates a signal 678 that is fed to multiplexer680. The multiplexer 676 also receives a signal 682 from the controller612. The multiplexer 680 generates a signal 684 that is received bycomponent 686 that is a MATLAB data file where all resulting simulationdata results are stored and from which all graphs shown in FIGS. 6-12,14-17 and 20 are extracted

5.1 Simulations of Following Errors

In what follows, the various feedforward controls are sequentiallyswitched in. To compare the effects, all simulation parameters were keptconstant.

System Parameters:

Momentary constant Ktc[Nm/A]=1.5*sqrt(2)

Momentary load inertia J_(L)[kgm²]=9

Rpm losses Nm/ω=0.15

Control device circuit parameters:

Position control device amplification MP1510 m/min/mm=9

P-factor (speed control device) MP2500 [As]=9

I-factor (speed control device) MP2510 [A]=2200

Interpolation parameters:

-   Jerk r [m/s³]2*10³-   Acceleration a [m/s²]=5-   Speed v [m/s]=0.4/60-   Position s [m]=4*10⁻⁴.    5.1.1 Following Error without Feedforward Control

The resulting following error without feedforward controls isrepresented in FIG. 14. A maximum following error of approximately 45 μmresults, which is impermissibly high.

5.1.2 Following Error with Speed Feedforward Control

The resulting following error without feedforward controls isrepresented in FIG. 15. A maximum following error during theacceleration phase of 10 μm results.

5.1.3 Following Error with Acceleration Feedforward Control

The resulting following error with acceleration feedforward control isrepresented in FIG. 16. As can be seen, no following error can be shown.

5.1.4 Following Error with IPC (Without IPC Feedforward Control)

The resulting following error with feedforward control and IPC (withoutIPC feedforward control) is represented in FIG. 17. As can be seen, afollowing error of 13 μm is built up at the end of the accelerationphase.

5.2 Installation of an IPC Feedforward Control into the Speed ControlDevice

To reduce the following error during the acceleration phase it isnecessary to implement an acceleration feedforward control. Since theinput value of the IPC is a speed, a multiplication of the accelerationfeedforward control a_soll(ipo) with the time constant T1 is necessary.

To make possible an implementation with optimized computing time, thefeedforward control summing point was moved ahead from the controldevice output to the IPC input, the structure represented in FIG. 18results in the process, i.e. IPC with acceleration feedforward control.

As shown in FIG. 18, a resultant signal 700 is formed as the combinationof the product of signals a_soll and T₁ being added to the signaln_soll. The resultant signal 700 is fed to the IPC 702 which generates asignal 704 that is added with the signal n_ist so as to form a signal706. The signal 706 is then fed to an integral. branch 708.

A further correction of the following error can be achieved by means ofa jerk feedforward control. The feedforward control value “r_soll(ipo)”can be formed in the speed control device by simple differentiation of“a_soll(ipo)”. The time error of half a scanning time occurring in theprocess only plays a subordinate role.

The IPC with acceleration and jerk control is represented in FIG. 19. Asshown in FIG. 19, a resultant signal 800 is formed as the combination ofthe signals r_soll, Tr, a_soll, T₁ and n_soll so that resultant signal800 is fed to the IPC 802 which generates a signal 804 that is addedwith the signal n_ist so as to form a signal 806. The signal 806 is thenfed to an integral branch 808 which in turn generates an output signal810. The proportional branch 812 receives a signal 814 so as to generatean output signal 816 that is added with the output signal 810.

5.2.1 Following Error with Convent. Feedforward Control, IPC and IPCPilot Control

In the simulation the relevant feedforward controls were expanded withthe above structure and compared with a structure wherein thefeedforward control point is located at the control device output. Nodifferences resulted here.

The resulting following error with conventional feedforward control, IPCand IPC feedforward control is represented in FIG. 20.

If the IPC feedforward control branch is installed, there is again nodetectable following error.

The structure of the speed control device block with feedforward controlin the control device output is represented in FIG. 21. In particular,the structure includes six input signals 900, 902, 904, 906, 908, 910.The input signal 900 is fed to a switch 912. The input signal 906 is fedto an amplifier 914 where the amplified signal 916 is fed to an IPCmodel feedforward control 918 that applies the factor 1/MP2500. Theresultant signal 920 is fed to switch 912. The switch 912 also receivesa constant signal 922.

The signal 908 is fed to the IPC phase reference model control 924 thatalso applies the factor 1/MP2500 so as to generate signal 926 that isfed to switch 928. The switch 928 also receives signal 908 and signal930. The switch 928 chooses one of the three signals 908, 926 and 930and feeds them to an adder 932 that also receives signal 910. The addedsignal 934 is sent to a component 936 that applies the factor P2510/2*sso as to generate signal 938.

As shown in FIG. 21, the signal from the switch 912 and the signals 902and 922 are sent to a switch 940 that sends one of the three signals toboth adder 942 and adder 944. The adder 942 receives the signal fromswitch 940 and signal 938 and adds the two to generate signal 944 whichis sent to multiplexer 946. The multiplexer 946 also receives signals956 and 965 and sends a signal 966 to an output.

Signals 902, 922 and 948, which is the result of the amplification ofsignal 904 by amplifier 950 are sent to switch 952 where one of thethree is sent to adder 944. The adder 944 receives two other signals 954and 956. Signal 954 is the result of amplifying signal 910 by amplifier958. Similarly, signal 956 is the result of amplifying signal 960 viaamplifier 962. Signal 956 is the result of adding signals 908 and 910 byadder 964. As shown in FIG. 21, the signals from the switches 940, 952and signals 938, 954 and 956 are combined by adder 944 to generatesignal 965 that is sent to an output.

6. Practical Examination of the IPC

The practical examinations were performed on a DIGMA 700. Initially, a1st order IPC, as had been employed in the above simulation, wasimplemented in the DSP software. Only small advantages result here whenthe IPC is used, the position control kv could only be increased byapproximately 15%.

It was therefore necessary to use an IPC of higher order, which bettercorresponds to the real system conditions.

6.1 Use of a 2nd Order IPC

An implementation of the 2nd order IPC was used after the followingconversion function:H(s)=1/(1+(T ₁ *s)+(T ₂ *s ²))

This is the conversion function of a PT2 capable of oscillation withdamping D.D=α/β=T ₁/(2*T ₂)

A damped oscillation is to be expected in actual machine tools.Therefore, damping D moves in the range 0<D<1.

The time constant T2 is calculated as follows:T ₂ =T ₁/(2*D)

Clearly improved results were already achieved with the use of a 2ndorder IPC, however, they still did not approach the results of thesimulation, which lead to conclusions of a theoretical increase of theposition control amplification kv of approximately 170%. The followingtime constants were determined for the DIGMA 700:

Time Constants at DIGMA 700:

X-axis Y-axis Z-axis MP25 15 15 12 MP26  0.0212  0.0205  0.0165 T1′  1 1  1 T2′  0.0017  0.0018  0.0018 T1  1.41 ms  1.37 ms  1.37 ms T2  1.7ms  1.8 ms  1.8 ms D  0.41  0.39  0.38

The below table shows the position control amplifications (kV factors)achieved at the X-axis of the DIGMA 700 in connection with various IPCdesigns. A search for the oscillation threshold was always performedhere. In accordance with a rule of thumb, the latter must always bemultiplied by a factor of 0.65 for stable operations.

kV (Oscillation limit) kV (stable) Without IPC 8.5 5.5 1st order IPC 9.56.2 2nd order IPC (D = 0.5) 13.0 8.5 2nd order IPC (D = 0.41) 14.5 9.5

Thus, the position control amplification could be increased to 170%.

6.2 Derivation of the IPC Algorithm

The derivation of the IPC algorithm is based on the equation:

 H(s)=1/(1+(T 1*s)+(T 2 ² *s ²)

Determination of the T2 Time Constant

Tests with DIGMA 700 have shown that the T2 time constant, and thereforedamping, is optimally set when the following error in the jerk phaseshowed a minimal deviation (with integrated jerk feedforward control).It was possible in this way to determine the T2 time constant for allthree axes.

In the lab set-up (JL directly on the motor shaft) it was also possibleto perform the determination of the optimum T2 time constant in thisway.

Connection between Damping and T2 Time ConstantD=T ₁/(2*T ₂)6.3 Employment in Machines with Dominant Natural Frequency

A further employment option of the IPC is provided when in connectionwith machines with low natural resonance and insufficient damping theIPC time constants are matched to the controlled system.

In connection with first tests performed during production on the“Chiron FZ 22L” it was possible to increase the kV factor from 1 to 5.However, it was not possible in this case to use the time constant T1determined from MP26 and MP25. It was necessary to employ a considerablyhigher time constant (approximately factor 5), which compensates a timeconstant in the machine.

In addition to “Chiron FZ 22L” a second machine, a Deckel-Mahon “DMU 50V” was tested.

The Deckel-Mahon “DMU 50 V” machine has strong resonances at 42 Hz and50 Hz. These are so dominant that it is only possible to set a jerk of10 and an acceleration of 1.5 at kV=4. By means of the use of the IPC itwas possible to achieve a kV of 12 for all axes. The values for jerkcould be increased to 20, acceleration was raised to 3.

Time Constants at DMU 50 V:

X-axis Y-axis Z-axis MP25 15 4.8 5.4 MP26  0.045 0.016 0.016 T1′  0.00420.0052 0.0052 T2′  0.003 0.0022 0.0013 T1  4.2 ms 5.2 ms 5.2 ms T2  3.0ms 2.2 ms 1.3 ms D  0.70 1.18 2.00

The speed control device settings were not changed (originalDeckel-Maho).

Result: A noticeable improvement in the position control device behaviorcould be achieved with both machines by the use of the IPC.

7. IPC Adjustment

When using the IPC it is necessary to differentiate between two types ofmachines. Type 1 is a rigid machine of not too large structural size,which is mostly directly driven or has linear motors. Type 2 is amachine with a dominant natural frequency in the range between 15 Hz to80 Hz, in which no sufficiently large kV factor can be set.

7.1 Adjustment of Rigid Machines

With machines of the type 1 it is sufficient as a rule if the IPC isswitched on with T1′=1 and T2′=0. The kV factor is increased until anoticeable oscillating tendency is noticed in the process.

Once this kV factor has been found, a fine adjustment of the IPC timeconstant T2 takes place. To this end first a T2 starting value ofT 2=2/3*MP 26/MP 25is set. Thereafter T2′ is changed until a new maximum kV factor has beenfound. Usually the T2 time constant must be reduced with this machinetype (down to maximally 0.5×the starting value). However, an increasewith respect to the starting value is also conceivable.

At the end the kV factor for the oscillation threshold must bemultiplied by the factor 0.65 in order to assure a sufficient stabilityof the position control device.

With this type of machine an increase of the kV by a factor of 1.4 to1.7 is possible.

7.2 Adjustment of Machines with Dominant Natural Frequencies

With machines of the type 2, the same adjustment should initially beperformed as with machine of the type 1. The IPC must be switched onwith T1′=1, and it is necessary to determine T2. In this case it is alsopossible that a T2 time constant results which is clearly greater thanthe T2 starting value.

Now the T1 time constant must be determined. For this purpose a T1starting value must be entered into MP2602 in place of a 1. It iscalculated fromT 1=MP 26/MP 25

This starting value must be increased until a maximum kV factor has beenfound. If the found T1 time constant is clearly greater than thestarting value (>factor 2), another adjustment of the T2 time componentshould take place. The value so far found should be lower, or raised,during testing.

Finally, the kV factor for the oscillation threshold must be multipliedby the factor 0.65 in order to assure a sufficient stability of theposition control device.

With machines of the type 2 a greater increase of the kV than by thefactor 1.7 is possible.

The foregoing description is provided to illustrate the invention, andis not to be construed as a limitation. Numerous additions,substitutions and other changes can be made to the invention withoutdeparting from its scope as set forth in the appended claims.

1. A method for determining at least one time constant of a referencemodel, which is designed as a 2nd order time-delay element of a machine,said method comprising: detecting an oscillation frequency of anundamped machine oscillation; and determining an optimized value of atime constant of said reference model as a function of said detectedoscillation frequency of said undamped machine oscillation, wherein saidreference model is arranged in a cascaded control arrangement and islocated between a position control device with a loop gain and a closedspeed control device, which comprises a proportional branch and anintegral branch, and wherein said reference model at least essentiallysimulates the behavior of said closed speed control circuit withouttaking said integral portion into consideration.
 2. The method of claim1, comprising: presetting a starting value of said time constant;presetting a starting value of a second time constant of said referencemodel; and increasing a loop gain of said position control device insteps up to a first maximum loop gain, at which an undamped machineoscillation is registered.
 3. The method of claim 2, wherein saidstarting value of said time constant is preset to zero and said startingvalue of said second time constant is preset to zero.
 4. The method ofclaim 2, wherein said optimized value is determined in accordance withthe equation:T 2 _(—) OPT=f(f _(S1))=1/(2*π*f _(s1)), wherein f _(s1)=saidoscillation frequency.
 5. The method of claim 2, wherein said secondtime constant is determined from preset system parameters.
 6. The methodof claim 5, wherein said second time constant is determined inaccordance with the equation:T 1 _(—) OPT=(J _(L)*2*π)/(k _(p) *K _(MC)) wherein J_(L): Momentaryload, k_(p): Loop gain of the proportional branch of the speed controldevice, K_(MC): Motor constant.
 7. The method of claim 5, furthercomprising checking whether said previously determined time constantassures a desired control behavior of said position control device. 8.The method of claim 7, wherein said increasing of said loop gain isaccomplished by using said optimized time constant, until an undampedmachine oscillation is registered, and an associated loop gain is usedas a second maximum loop gain during subsequent operation of saidmethod.
 9. The method of claim 8, further comprising multiplying saidsecond maximum loop gain by a safety factor K, wherein K<1.
 10. Themethod of claim 5, further comprising checking whether said second timeconstant provides an acceptable system behavior, or whether anoptimization of said second time constant must be performed.
 11. Themethod of claim 10, further comprising optimizing said second timeconstant by, proceeding from said starting value for said second timeconstant, changing said second time constant in steps until saidundamped machine oscillation is registered, and a value of saidoptimized second time constant obtained therefrom is used as anoptimized value for parameterizing said reference model.
 12. The methodof claim 11, further comprising: using said optimized time constant andsaid second time constant; and increasing said loop gain until anundamped machine oscillation is registered, and using an associated loopgain as a second maximum loop gain in subsequent operation of saidmethod.
 13. The method of claim 1, wherein said optimized value isdetermined in accordance with the equation:T 2 _(—) OPT=f(f _(S1))=1/(2*π*f _(s1)), wherein f _(s1)=saidoscillation frequency.
 14. The method of claim 1, wherein said method isexercised in an automated manner.
 15. The method of claim 14, whereinsaid machine theoretically requires an nth order reference model,wherein n>2 applies.
 16. The method of claim 1, further comprising usingin said machine said reference model with said optimized value of saidtime constant.
 17. A method for determining at least one time constantof a reference model, which is designed as a 2nd order time-delayelement of a machine, said method comprising: detecting an oscillationfrequency of an undamped machine oscillation; and determining anoptimized value of a time constant of said reference model as a functionof said detected oscillation frequency of said undamped machineoscillation, wherein said optimized value is determined in accordancewith the equation:T 2 _(—) OPT=f(f _(S1))=1/(2*π*f _(s1)), wherein f _(s1)=saidoscillation frequency.
 18. A method for determining at least one timeconstant of a reference model, which is designed as a 2nd ordertime-delay element of a machine, said method comprising: detecting anoscillation frequency of an undamped machine oscillation, anddetermining an optimized value of a time constant of said referencemodel as a function of said detected oscillation frequency of saidundamped machine oscillation, wherein said machine theoreticallyrequires an nth order reference model, wherein n>2 applies and saidmethod is exercised in an automated manner.
 19. The method of claim 18,further comprising using in said machine said reference model with saidoptimized value of said time constant.
 20. A device for determining atleast one time constant of a reference model, which is designed as a 2ndorder time-delay element of a machine, said device comprising: areference model arranged in a cascaded control arrangement; a positioncontrol device with a loop gain; a closed speed control device, whichcomprises a proportional branch and an integral branch and wherein saidreference model is located between said position control device and saidclosed speed control device; a detector for detecting an oscillationfrequency of an undamped machine oscillation; and an optimizer thatdetermines an optimized value of a time constant of said reference modelas a function of said detected oscillation frequency of said undampedmachine oscillation.
 21. The device of claim 20, wherein said referencemodel at least essentially simulates the behavior of said closed speedcontrol circuit without taking said integral portion into consideration.